What is the significance of cos²α+cos²β+cos²γ = 1 in crystallography?

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The equation cos²α + cos²β + cos²γ = 1 is significant in crystallography as it relates to the projection of a unit vector in three-dimensional space. When angles α, β, and γ are defined with respect to the x, y, and z axes, this equation holds true, reflecting the Pythagorean theorem in three dimensions. The components of the vector correspond to the cosine of these angles, confirming that the sum of their squares equals one. In two dimensions, a similar relationship emerges, where cos²α + cos²β simplifies to 1 due to the complementary nature of the angles. This illustrates the foundational geometric principles underlying crystallographic structures.
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On pg 6 of http://www.scribd.com/doc/3914281/Crystal-Structure, it quotes this result without proof. My notes from uni also quote this result but I can't see where it comes from. Does anyone know? Thanks
 
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It looks like the Pythagorean theorem applied in three dimensions.
 


Well, it isn't true for just any angle \alpha, \beta, \gamma, of course! If, however, you draw a line through the origin of a three dimensional coordinate system, define \alpha to be the angle the line makes with the x-axis, \beta to be the angle the line makes with the y-axis, and \gamma to be the angle the line makes with the z-axis, then this is true.

To see that, think of a vector of unit length in the direction of that line. If we drop a perpendicular from the tip of the vector to the x-axis, we have a right triangle in which an angle is \alpha and the hypotenuse is 1, the length of the vector. Thus, the projection of the vector on the x-axis, and so the x-component of the vector is cos(\alpha). Similarly, the y-component of the vector is cos(\beta) and the z-component of the vector is cos(\gamma). That is, cos^2(\alpha)+ cos^2(\beta)+ cos^2(\gamma) is the square of the length of the vector which is, of course, 1.

By the way, look what happens if you do this in two dimensions. If you have a line in the plane through the origin making angles \alpha with the x-axis and \beta with the y- axis then \beta= \pi/2- \alpha so cos^2(\alpha)+ cos^2(\beta)= cos^2(\alpha)+ cos^2(\pi/2- \alpha)= cos^2(\alpha)+ sin^2(\alpha)= 1.
 


That was embarrassing.. thanks
 
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